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Unformatted text preview: COMPACTIFICATIONS OF MODULI OF ABELIAN VARIETIES: AN INTRODUCTION. MARTIN OLSSON Abstract. In this expository paper, we survey the various approaches to compactifying moduli stacks of polarized abelian varieties. To motivate the different approaches to com- pactifying, we first discuss three different points of view of the moduli stacks themselves. Then we explain how each point of view leads to a different compactification. Through- out we emphasize maximal degenerations which capture much of the essence of the theory without many of the technicalities. 1. Introduction A central theme in modern algebraic geometry is to study the degenerations of algebraic varieties, and its relationship with compactifications of moduli stacks. The standard example considered in this context is the moduli stack M g of genus g curves (where g 2) and the Deligne-Mumford compactification M g M g [9]. The stack M g has many wonderful properties: (1) It has a moduli interpretation as the moduli stack of stable genus g curves. (2) The stack M g is smooth. (3) The inclusion M g , M g is a dense open immersion and M g \ M g is a divisor with normal crossings in M g . Unfortunately the story of the compactification M g M g is not reflective of the general sit- uation. There are very few known instances where one has a moduli stack M classifying some kind of algebraic varieties and a compactification M M with the above three properties. After studying moduli of curves, perhaps to next natural example to consider is the moduli stack A g of principally polarized abelian varieties of a fixed dimension g . Already here the story becomes much more complicated, though work of several people has led to a compact- ification A g A g which enjoys the following properties: (1) The stack A g is the solution to a natural moduli problem. (2 ) The stack A g has only toric singularities. (3 ) The inclusion A g , A g is a dense open immersion, and the complement A g \ A g defines an fs-log structure M A g (in the sense of Fontaine and Illusie [16]) on A g such that ( A g ,M A g ) is log smooth over Spec( Z ). Our aim in this paper is to give an overview of the various approaches to compactifying A g , and to outline the story of the canonical compactification A g , A g . In addition, we also consider higher degree polarizations. 1 2 MARTIN OLSSON What one considers a natural compactification of A g depends to a large extent on ones view of A g itself. There are three basic points of view of this moduli stack (which of course are all closely related): ( The standard approach ). Here one views A g as classifying pairs ( A, ), where A is an abelian variety of dimension g and : A A t is an isomorphism between A and its dual (a principal polarization ), such that is equal to the map defined by an ample line bundle, but one does not fix such a line bundle. This point of view is the algebraic approach most closely tied to Hodge theory....
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